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A079216
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Square array A(n>=0,k>=1) (listed antidiagonally: A(0,1)=1, A(1,1)=1, A(0,2)=1, A(2,1)=2, A(1,2)=1, A(0,3)=1, A(3,1)=3, ...) giving the number of n-edge general plane trees fixed by k-fold application of Catalan Automorphisms A057511/A057512 (Deep rotation of general parenthesizations/plane trees).
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14
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1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 5, 2, 1, 1, 6, 11, 3, 2, 1, 1, 10, 26, 8, 5, 2, 1, 1, 11, 66, 18, 11, 3, 2, 1, 1, 18, 161, 43, 30, 5, 5, 2, 1, 1, 21, 420, 104, 82, 6, 14, 3, 2, 1, 1, 34, 1093, 273, 233, 15, 38, 5, 5, 2, 1, 1, 35, 2916, 702, 680, 36, 111, 6, 11, 3, 2, 1, 1, 68, 7819, 1870
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Note: the counts given here are inclusive, e.g. A(n,6) includes the counts A(n,3) and A(n,2) which in turn both include A(n,1).
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LINKS
| A. Karttunen, Catalan Automorphisms
Index entries for sequences related to parenthesizing
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FORMULA
| A(0, k) = 1. A(n, k) = Sum_{r=1..n where r/gcd(r, k) divides n} Sum_{c as each composition of n/(r/gcd(r, k)) into gcd(r, k) parts} Product_{i as each composant of c} A(i-1, lcm(r, k))
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MAPLE
| with(combinat, composition); # composition(n, k) gives ordered partitions of integer n into k parts.
[seq(A079216(n), n=0..119)]; A079216 := n -> A079216bi(A025581(n), A002262(n)+1);
A079216bi := proc(n, k) option remember; local r; if(0 = n) then RETURN(1); else RETURN(add(PFixedByA057511(n, k, r), r=1..n)); fi; end;
PFixedByA057511 := proc(n, k, r) option remember; local ncycles, cyclen, i, c; ncycles := igcd(r, k); cyclen := r/ncycles; if(0 <> (n mod cyclen)) then RETURN(0); else add(mul(A079216bi(i-1, ilcm(r, k)), i=c), c=composition(n/cyclen, ncycles)); fi; end;
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CROSSREFS
| A(n, A003418(n)) = A000108(n). The first row: A057546, second: A079223, third: A079224, fourth: A079225, fifth: A079226, sixth: A079227. Cf. also A079217-A079222.
Sequence in context: A098805 A168396 A049286 * A181654 A192710 A112380
Adjacent sequences: A079213 A079214 A079215 * A079217 A079218 A079219
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KEYWORD
| nonn,tabl
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AUTHOR
| Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com) Jan 03 2002
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